B ULLETIN DE LA S. M. F.
D AN B URGHELEA
R EINHARD S CHULTZ
On the semisimple degree of symmetry
Bulletin de la S. M. F., tome 103 (1975), p. 433-440
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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
103, 1975, p. 433-440.
ON THE SEMISIMPLE DEGREE OF SYMMETRY
BY
DAN BURGHELEA and REINHARD SCHULTZ (1) [Bucuresti], [Lafayette, Ind.]
ABSTRACT. - One defines the semisimple degree of symetry Ss (M) for a compact manifolds M", as the heighest dimension of all compact semisimple Lie groups acting effectively on M"; one recognise the manifolds M" with small 5', (M") in terms of the cup- length.
RESUME. — On introduit Ie degre semisimple de symetrie Ss (M) d'une variete compacte M, comme la plus grande dimension des groupes de Lie compacts et semisimples qui agissent effectivement sur M"; on donne des conditions en termes de longueur des cup- produits permettant de majorer Ss (M).
For any compact topological (differentiable) manifold one defines the topological (differentiable) degree of symmetry as in [3] to be S (M) = sup { dim G', G compact Lie group acting topologically and effec- tively on M"} (resp. 5^ (M)), and the semisimple degree of symmetry is similarly defined: 8s (M) = sup { dim G; G compact semi simple Lie group acting topologically and effectively on M ] (resp., S^ (M)).
If AT is differentiable, then S^ (M") ^ S8 (M") and ^ (M") ^ S (M"). By [7] (p. 243), (^^(M^^M^^+l)/! These numbers are further related as follows:
PROPOSITION 1. - S (M") - 3s (M") ^ n (resp. ^ (M") - S^ (M") ^ n\
and if S^M^-S^M") = n then M" is homeomorphic (dijfeomorphic) to the torus T".
0 This paper was writen while the first named author was visiting at University of Chicago and Universite de Paris-VII; and the second named author was partially supported by National Science Foundation Grants GP-19530A2 and MPS74-03609.
434 D. BURGHELEA AND R. SCHULTZ
This is a consequence of the following three observations:
1° Every compact connected Lie group has a finite cover of the form Sx T^ with S semisimple.
2° If a group acts almost effectively, so does every subgroup.
3° It 7^ acts almost effectively on a connected manifold M, the principal orbit theorem implies that almost all orbits are ^-dimensional and (hence) dim M ^ k.
The following facts suggest some differential-geometric interest in the computation of S8 (AT):
(a) If M" is a differentiable manifold with S^ (M") = 0 then for any riemannian metric on M" two infinitesimal isometries X, Vhave [X, Y] = 0 (for the connected component of the isometry group is a torus).
(b) If S^ (M") ^ 0, M" admits a riemannian metric with positive scalar curvature.
Part (a) is obvious, and Part (b) is due to Lawson and Yau [5]. The purpose of this note is to give simple criteria in terms of rational (or real) cup length to estimate 8s (M"), and in particular recognize manifolds with S8 (M") = 0. These criteria are furnished by proposition 2. The cup length criteria parallel those of [3] (theorem 2), but the results obtained here are somewhat different.
DEFINITION. - We say M" has abelian symmetry if S55 (M") = 0, and AT has strong abelian symmetry if 8s (M") = 0 and for any effective action of a compact Lie group the isotropy groups are finite.
THEOREM A. — Let Mn be a compact connected manifold and suppose there exist W^, ..., W^ e H1 (M; R) with W^ u . . . u W^ ^ 0. Then
S8 (M") = 0; moreover M" has strong abelian symmetry.
Theorem A generalizes some results of S.-T. Yau [9], who proved the result for smooth actions (using harmonic forms) and for topological S1
actions having fixed points (using the Gysin-Smith sequence [2]). Proofs of Theorem A have also been independently obtained by J.-P. BOURGUI- GNON and F. RAYMOND.
Example. - T" # M" (M" any orientable manifold) has strong abelian symmetry. Thus n even and ^ (M # T") ^ 0 imply S (M # T") = 0.
THEOREM B. — Let M" be a compact connected manifold and suppose there exist W^, . . . , W^^ e H1 (M; R) and C e H2 (M; R) with W^
W1 u . . . u M^_2 u C ^ 0 and
(*) ^*(M; R) ^ Jf*(51; 2^) ®B, f = 2,3, H^/? 2? a Poincare duality cohomology algebra. Then S " (M") = 0.
In the differentiable category we can replace (*) by
(**) At least one rational characteristic number is nonzero.
This also true for locally smooth actions [2], but we shall not prove this generalization.
Example. - Tn~2x S2# Mn with Mn an oriented manifold and H * (Mn;R) ^ H^ {Sn;R) obviously satifies the hypotheses and has abelian symmetry but not necessarily strong abelian symmetry. For example Tn~2x S2 # T""2 x 5'2 carries an effective S ^action with nonempty fixed point set. To construct this action, let S1 act on Tn~2x S2 via I x u , where ^ is the standard linear action of S1 = SO^ on S2; since the fixed point set of this actions is T"~2 x { ± N ] , where N is the north pole of S2, we can form the equi variant connected sum of two copies of this action at (1, N). This gives a smooth ^-action on T"~2 x S2 # T"~2 x S2 which has fixed points.
THEOREM C. — Let -M" be a compact connected manifold. Furthermore, assume % (M") is odd and there exists W^y .. .,Wn-3 e H1 (M; R) and C e H3 (M; R) with W^ u . . . u ^.3 u C ^ 0. Then 8s (M) = 0.
Proofs. — The proofs of A, B, and C, are pleasant consequences of the following :
PROPOSITION 2 : [(^) Let [i : S1 x M" -> M" be a nontrivial action on a connected manifold and suppose that for any x e M",
a, .•^-^(^^c:^
induces the trivial homomorphism H1 (M; R) —> H1 (S1; R). Then for any W^, . . . , W^ e H1 (M; R), we have W^ u . . . u W^ = 0.
(V) Let G be a compact semisimple Lie group and [i: Gx M" -> M"
be an action on the connected manifold M". Let k = dim G/HQ where HQ is the minimal isotropy group. Assume that, for any point x e M", o^ : G/G^ -> M induces the trivial homomorphism H^ (M; R) -> H^ (G/G^; R) then, for any W^, .... W^ e H1 (M; R) and C e H1' (M; R), we have
W^u . . . u T ^ _ f e U C = 0 .
436 D. BURGHELEA AND R. SCHULTZ
COROLLARY 3. - The conclusions of proposition 2 hold if, for one x, the map ^ : G/G^ -> \i (G, x) c M induces the trivial homomorphism a; '.H^M; R)-.Hk(G|G,; R). In particular, this is true if ^ has orbits of different dimensions.
The corollary follows immediately using the connectedness of M, the principal orbit theorem, and the existence of slices (see [I], [2]).
We shall first derive theorems A-C from proposition 2 and corollary 3, after which we shall prove proposition 2.
Proof of theorem A. — If G is a compact semisimple Lie group, and [i: GxM" -^ M" is sin action, then p, restricted to any S1 x M" -> M" (S1 is a subgroup in G) is trivial by proposition 2; for the homomorphism
^'.H^M; R^H1^1; R) factors through H1 (G; R), which is zero.
Furthermore, if G is a torus then all orbits have the same dimension by corollary 3, and hence all isotropy subgroups must be finite if G is effective.
Proof of theorem B. - Because any compact semisimple Lie group contains a 3-dimensional compact semisimple Lie subgroup, it is enough to prove that any [i: S3xMn -> M" is trivial provided Mn satisfies the hypotheses of theorem B.
If the action has at least two orbits of different dimension, then by pro- position 2 the action has to be trivial. So assume that all the orbits of the action p : S3 xM" -» M" are 2-dimensional. If o^ : S3/ S3 -> M"
induces the trivial homomorphism o^ : H2 (M"; R) -> H2 (S3/S3,; R) the action is trivial by proposition 2. If not a^ : H* (M"; R) -> H * ( S3^ S3' , R) is surjective for all x, and every isotropy subgroup is conjugate to S1. In this case, M -> M/S3 is a sphere bundle whose fiber is totally nonhomolo- gous to zero. Therefore
^*(M; R) = (53/^3 ; R) ®^*(M/53; R) == H^(S2; R) ®^(M/53; R).
which by (*) is not possible. Also, M bounds MX go D3, which contra- dicts (**). In the case when all orbits have dimension 3, a similar ana- lysis (2) shows
Jf*(M; R) = H3^3; R) ®^*(M/53; R)
contradicting (*). If M is differentiable, since M/S3 is suitably triangu-
(2) Although M WMIS3 is not a fiber bundle, it behaves in real cohomology as if it were one since the orbits are all real cohomology spheres.
lable it follows that the mapping cydinder of/: M -> M/S3 is a triangu- lated rationnal cohomology manifold with boundary M. Hence all ra- tionnal characteristic numbers of M must vanish; but thiscont radicts (**).
Hence [i must be trivial.
Proof of theorem C. — Let [i : S3 xM" -> M" be an action. If [i has a 3-dimensional orbit, then by the cup product hypothesis proposition 2 and corollary 3 all orbits must be 3-dimensional. But in this case, the Leray spectral sequence implies % (M) = ^ (M/S3) % (^3) = 0 as before;
since ^ (M) is odd, p, has no 3-dimensional orbits.
Since all orbits have dimension 0 or 2, the classification of subgroups of 5'3 implies that the S3 action reduces to an SO^ action, each orbit of which is either a fixed point, RP2, or 5'2. If [i is nontrivial, the orienta- bility of M" implies that S2 is the principal orbit type. On the other hand, if [i is nontrivial there must also be nonprincipal orbits; for otherwise, M would be an S 2 bundle over M/SO^, so that / (M) = % (M/SO^) ^ (S2) would be even.
Let F be the fixed point set of SO^, and let E be the union of the RP2 orbits. We claim that
^(E',Z,)=^F;Z,)=O,
and F contains no limit points of E (assuming ^ is nontrivial). If this is true, then of course
/(M) = x(M; Z,) = x(M, £uF; Z,);
on the other hand, (M, (E u F)) is a relative S2 bundle and hence
% (M, E u F; Z2) is even, a contradiction since % (M) is odd. Hence [i is trivial if the above claim is true.
If H is a smooth action, we may prove the claim as follows: By the prin- cipal orbit theorem the local representations of SO^ at points of F all have SO^ as their principal isotropy subgroup. Since the only such repre- sentation (up to equivalence) is the usual one on R3, it follows that every component of F has codimension 3 and F contains no limit points ofE. Sin- ce x (At") is nonzero, n must be even and n — 3 must be odd; hence % (F) = 0.
On the other hand, by the differentiable slice theorem, an RP2 orbit has an invariant tubular neighborhood of the form ^ x ^ V for some Z^-repre- sentation V\ let VQ be its fixed point set. Since RP2 is nonorientable but M" is orientable, dim F-dim VQ must be odd. It follows that E is also a
438 D. BURGHELEA AND R. SCHULTZ
union of closed submanifolds each having odd codimension (hence odd dimension), so that % (E) = 0.
A similar sort of argument applies to topological actions. The local analog of some results due to W.-Y. HSIANG ([4], p. 346-349) shows that F is a union of closed Z-cohomology (^-3)-manifolds, so that
X {F) = x (F; Z,) = 0
still holds. In addition, the proof of [4] (Proposition 3, p. 348) shows that F contains no limit points of E. The topological slice theorem implies that every RP2 orbit has a neighborhood of the form S2x^V for some Z-cohomology manifold V\ let VQ be its fixed point set. The cohomolo- gical characterization of orientability, and P.A. Smith theory again imply VQ is a Z^ — cohomology manifold and coh dim F-coh dim VQ is odd. Thus E is again a union of closed odd-dimensional Z^-cohomology manifolds, so that % (E; Z^) is still zero. This completes the proof of theorem C.
Proof of proposition 2:
(a) Looking at the Leray spectral sequence E^ => H * (M; R) associa- ted to/; M -^ M/S1 it is easy to see that any 1-class has filtration ^ 1 since H1 (M; R) -. H1 (S^S^; R) is trivial (because the edge map H1' -. E^ c Ey = r(Rkf^ R) sends a class w into the section S(x) = w\Hk (Gx)). Thus the long cup product has filtration ^ n. Since E^^ = 0 for p ^ n-1 +1 = n, the cup product clearly vanishes (See [8], XIII, for the relevant multiplicative properties).
(b) If G is compact semisimple, the Leray spectral sequence E^ has the line q = 1 consisting of trivial groups because H1 (G/G^; R) = 0. Hence 1-dimensional classes in H1 (M; R) have Leray spectral sequence filtration
^ 1. On the other hand, because H^ (M; R) -. H^ (G\G^; R) is zero every /^-dimensional class also has filtration ^ 1 (look again at the edge map Hk -> E0^ <= E^ = T(Rk(f^ R)). Thus the cup product has filtration ^n-k+l(n= dim M). Since E^^ = 0 for p ^ dim M/G+1 == n—k+1, the cup product under consideration clearly vanishes.
Addendum on Massey products. - In [9], YAU points out the following strengthening of proposition 2 (i): If 7^ acts effectively on M", and
H1 (M"; R) -> H1 (T^- R)
is trivial, then all higher order Massey products involving monomials in H1 (M"; R) are trivial provided the degrees of any two successive ones are
^ n -k+1. Our methods also yield this. For the strengthened hypo- thesis implies that E^1 = 0 in the Leray spectral sequence for AT -> Mn/Tk, and hence the edge map H1 (M"/!^; R) -, H1 (M"; R) is onto; but the corresponding Massey products in H * (M"/^/.^) are all defined and trivial since the latter cohomology vanishes above degree n — k = dim Mn/Tk. Of course, all Massey products in H " (M"; R) are trivial since the inde- terminacy is total by Poincare duality. However, it does not seem that further triviality conditions for Massey products are obtainable, even when such products are always defined. For example, Poincare duality and pro- position 2(i) imply that every product W^, . . . , W^-^ is zero, and hence the Massey product
< T ^ , W^ ..., W^ ^Oejr-^M";^)
is always defined. However, one can construct M" with a free S1 action and a nontrivial Massey product of the above sort as follows: Let n ^ 4, and choose generators ^i, . . . , ^-i,/i, .. .,/„-! of H1 (T"-1 #Tn~l; R) so that the e ' s come from the first T""1 summand, the/'s come from the second, and
o^n^-n^-iV.
Take M" to be the principal S1 bundle with Euler class/i/^, and let n :M" -^ T""1 # T""1
be the projection. Then the Massey product
<7l*^, 7T*(^, . . ., ^_,), 7l*/, ^iT-^M"; R)
is nontrivial; this follows easily from the fact that suitable Massey products in (£2, d^) of the Serre spectral sequence pass to Massey products in
H * (M"; R) (the argument of [6] (Theorem 4.1) is applicable).
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440 D. BURGHELEA AND R. SCHULTZ
[3] HSIANG (W.-Y.). — On the degree of symmetry and the structure of highly symmetric manifolds, Tamkang J. Math., t. 2, 1971, p. 1-22.
[4] HSIANG (W.-Y.). — On the splitting principle and the geometric weight systems of topological transformation groups, I., "Proceedings of the second conference on compact transformation groups [1971, Amherst]", vol. 1, p. 334-402. — Berlin, Springer-Verlag, 1972 (Lectures Notes in Mathematics, 298).
[5] LAWSON (H. B.) and YAU (S.-T.). - Scalar curvature, nonabelian group actions, and the degree of symmetry of exotic spheres. Comment. Math. Helvet., t. 49, 1974, p. 232-244.
[6] MAY (J. P.). - Matric Massey products, J. of Algebra, t. 12, 1969, p. 533-568.
[7] MONTGOMERY (D.) and ZIPPIN (L.). — Topological transformation groups. — New York, Interscience Publishers, 1955 (Inter science Tracts in pure and applied Mathe- matics, 1).
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[9] YAU (S.-T.). — Remarks on the group of isometrics of a riemannian manifold, SUNY at Stony Brook, 1974 (multigr.).
(Texte recu Ie 31 decembre 1974.) Dan BURGHELEA,
Institut de Mathematiques, 14, rue Academiei, Bucuresti (Roumanie)
et Reinhard SCHULTZ, Division of Mathematical Sciences
Purdue University, Lafayette, Indiana 47907,
(Etats-Unis).
